Month: November 2015

Sometimes I see a worksheet online and I say to myself, “That should stay a worksheet. Paper is the right home for that math. Any possible benefit from moving that math to a computer is more than outweighed by the hassle of dragging out the laptop cart.”

Other times I see a worksheet and it seems clear to me that a different medium would add – you name it – breadth, depth, interest, collaboration, etc.

That’s the case with Joshua Bowman’s implicit differentiation worksheet, which he shared on Twitter. It’s great in worksheet form. But the Desmos Activity Builder can add a lot here while subtracting very little. Activity Builder is the right home for this math.

Bowman is asking his students to do their work in Desmos anyway and then copy and paste their calculator link into a Google Doc for feedback.

Activity Builder simplifies that collection process. Students do their work in the Desmos activity. Desmos sends you all of their graphs, quickly clickable.

Ask More Questions

When students see worksheets with seventeen questions running (a) through (q), they lose their mind. Let’s lighten their cognitive load and keep question (q) out of their visual space while they’re considering question (a).

This isn’t necessarily an improvement, especially if my new questions just ask students to repeat the same dreary work several hundred times. So:

Ask More Interesting Questions

I added six more questions to Bowman’s worksheet, and they share particular features.

First, they ask students to work at several different levels, from informal to formal. For example, I wanted to ask questions about:

a blank graph – “What do you think the shape of the graph will be?”

the graph – “Add up all the intercepts. What is that sum?”

the graph and some tangent lines – “Multiply their slopes. What is the product?”

These questions move productively from informal understandings to formal understandings, but they don’t live well together on the same piece of paper. You can’t ask students, “What do you think the shape of the graph will be?” when the graph is farther down the page.

Another example:

Bowman’s worksheet asks students to find the equation of the tangent lines to the intercepts of the graph. Some students may use sliders, other students may differentiate implicitly.

I can quickly figure out which group is which by asking them to multiply their slopes together and enter the product in a new question. Which students differentiated and which students experimented?

Long before I ask students to calculate that product, I ask them to simply estimate its sign. Envision the tangent lines in your head. Without knowing their exact slopes, what will their product be? That’s an informal understanding that assists later, formal understandings.

So again:

Simplify assignment collection.

Ask more questions.

Ask more interesting questions.

Best of all, this Desmosification took minutes. Start somewhere. The tools are all free forever. Thanks, Joshua, for sharing your worksheet and letting us take a crack at it.

Here is my speaking calendar for 2016. Some of these sessions are private, others have open registration pages (see the links), and others have waiting lists. Feel free to send an e-mail to dan@mrmeyer.com with inquiries about any of them. It’d be a treat to see you at a workshop or a conference.

BTW. After my keynote address at Nebraska’s state conference on September 9, 2016, I’ll have worked with teachers in every U.S. state. It’s been such a privilege getting to know so many interesting people doing so much interesting work. If you have attended any of my sessions, you’ve heard me express how indebted I am to participants from other sessions for the questions they ask about my ideas and the ideas they share themselves.

We’re continuing to host commenters from across a vast philosophical divide (including the co-authors of The Atlantic article under discussion) commenters who are unlikely to share the same physical space any time soon. People have largely kept it together and you’d have to be a committed ideologue not to walk away with a better understanding of the people who disagree with you.

I haven’t been able to shake one particular exchange, though.

Halfway through the comments, two people who disagree with each other as completely as anyone could each made a precise and articulate case for their diametrically opposite theories of learning.

I thought the purpose of a problem in a classroom is to check whether a student knows sufficient math to solve it, rather than learn bout the nature of human thinking processes. If it is the latter, Dan is completely right, except it belongs to cognitive science experiment rather than a classroom.

Brett Gilland, an infrequent blog commenter who should comment more frequently:

I can not disagree with this enough. The purpose of a problem in my classroom is almost always to understand the nature of that human’s thinking processes. This allows for amplification, further investigation into how the student is able to navigate similar problems with subtle variations and complications, and attempts to draw student mental models into internal conflict to create pressure for remediation and revision of said mental models.

I suspect that Gilland’s employer, and certainly the parents of his students, would also disagree. Some quite strongly. The primary purpose of school is to educate the kids at hand, not to train the teacher. This doesn’t mean that teachers do not learn from experience, but if gaining experience and insight is the primary reason for what the teacher does, he’d better get approval from an IRB and a waiver from each individual student or parent that attend his class.

Funny thing, that. My employer, my parents, my students, my district, the state evaluator for my school, etc. all support my teaching. Most quite strongly. This might be due to the fact that when most people hear “I work really hard to understand your child’s thought processes so that I can better guide their thinking and draw out subtleties and conflicting mental models,” they don’t think “Oh my God, that man is performing experiments on my child to improve his educational practice.” Instead, they think “Oh my God, that man really cares about what is going on inside my child’s head and is attempting to tailor instruction to what he finds there. Thank goodness he isn’t stuck with a teacher who believes that teaching is just lectures interspersed with quizzes to determine if my child gets it or needs to be droned at more with another utterly useless generic explanation that takes no account of what my particular child is thinking!”

I’m sure that everyone walked away feeling like their side won, but one side is wrong about that.

Alas, I feel that Mathematics is reaching a junction – in which the traditionalists and the progressives must come to a head and work together to forge a stronger future for our young mathematicians! Whilst today’s world demands an ability to think and to use available resources to find new meaning, we must not forget those who generated those resources in the first place. a fine craftsman needs to learn the tools of his trade before he or she can produce the creative thinking in his head. A computer programmer must use efficient logic before we can play those game or use those apps to be progressive learners. To what degree should thinking, reasoning, and problem solving come before skills acquisition , or vice versa? Sound like the chicken and the egg to me.

My biggest fear for us as teachers is that we are robbing our young people of the beauty and passion for maths through repetitive, applied drill, just so that they can demonstrate a high level skill skill that they will never use. or, we are not providing enough technical skills to enough of our students to ensure quality craftsmanship I am saddened every single time that my best mathematics students tell me they want to study medicine or dentistry instead of using their mathematical ability to grow our field.

Let us first and foremost provide our students with mathematical challenge- that requires both the creative solution and advanced skills acquisition. Whether the challenge be abstract or modelling a real situation need not matter. What is important is bringing back the passion for mathematics that we, as mathematics educators share, passion- through intrinsic motivational challenge and drive.

Their article: “At best, verbal explanations beyond ‘showing the work’ may be superfluous; at worst, they shortchange certain students and encumber the mathematics for everyone.”

I can see why The Atlantic would want to sharpen their writing for the headline. They qualify themselves twice in the article (“at best” and “may be”) barely making a claim.

So if they think symbols are always sufficient explanations, let’s offer questions in the comments for which they aren’t. If they think verbal explanations are sometimes necessary, let’s let them articulate when.

There’s a place in instruction (somewhere between ages 3 and 8) where each of the symbols “3” and “+” and “4” and “=” and “7” each need explanation, which might look like
… + …. = … …. = …….

I am pretty sure that Common Core haters dislike the notion that any of that ought to be explained, that they would prefer that this just be one of the 55 addition facts that ought to be memorized, and let’s move on.

At the same time, requiring 3rd or 4th graders to explain why 3 + 4 = 7 seems ridiculous, EVEN THOUGH SOME 3RD GRADERS HAVEN’T MASTERED ADDITION.

I would ask my 8th graders to explain why 3x + 4 != 7x, but I wouldn’t ask for this in a Calculus class, EVEN THOUGH IT HAS NOT BEEN MASTERED BY ALL.

My point is that we use symbols for efficiency, to avoid explanation. Symbols are NEVER enough explanation at the beginning of instruction.

A student’s presence in a given class presumes a level of previous mastery and efficiency WHICH IS NOT ALWAYS THERE, and instruction examples demonstrate the level of explanation that is expected.

To Chester Draws about the quadratic, I would hope for words like “Quadratic => 0, 1, or 2 solutions” in an explanation.

So a really good question is “what level of explanation should students be expected to demonstrate on a national test? (all right, multi-state, but I am in favor of a national curriculum).

Is it really the case that the non-linguistically inclined student who progresses through math with correct but unexplained answers—from multi-digit arithmetic through to multi-variable calculus—doesn’t understand the underlying math? Or that the mathematician with the Asperger’s personality, doing things headily but not orally, is advancing the frontiers of his field in a zombie-like stupor?

I wouldn’t bet that the student with correct but unexplained answers understands nothing, but I wouldn’t make any confident bets on exactly what that student understands either.

Math answers aren’t math understanding any more than the destination of your car trip indicates the route you took. When five people arrive at the same destination, asking how each arrived tells you vastly more about the city, its traffic patterns, and the drivers, than just knowing they arrived.

Their other exemplar of understanding-without-explaining is strange also. Mathematicians advance the frontiers of their field exactly by explaining their answers – in colloquia, in proofs, in journals. Those proofs are some of the most rigorous and exacting explanations you’ll find in any field.

Those explanations aren’t formulaic, though. Mathematicians don’t restrict their explanations to fragile boxes, columns, and rubrics. Beals and Garelick have a valid point that teachers and schools often constrain the function (understanding) to form (boxes, columns, and rubrics). When students are forced to contort explanations to simple problems into complicated graphic organizers, like the one below from their article, we’ve lost our way.

Understanding is the goal. The answer, and even the algebraic work, only approximate that goal. (Does the student know what “80” means in the problem, for example? I have no idea.) Let’s be inflexible in the goal but flexible about the many developmentally appropriate ways students can meet it.

Yet another important thing about students explaining their reasoning is that there is great self-help in a careful explanation of processes. How often have we had a student explain a problem he/she did incorrectly and, in the explanation, the student realizes the mistake without a word from us? This “out-loud-silently-in-my-head” thinking is such an important thing to help students develop.

Another reason that we might want to listen or read a student explanation of how they solved a problem is just so, in the process of articulating their solution, students may run into their own inconsistencies in their work. I have noticed, quite often, that students will give an answer that I don’t understand, and then when I ask them to explain what they did, in the middle of their explanation they say something like, “Oh, oops! Yeah that isn’t right. I mean this instead” and revise their thinking.

Mathematicians use words in describing their discoveries all the time – and have for a long time. That’s why some doctorates in mathematics require a foreign language so that the candidate can read the mathematicians’ writings in the original language.

1) Can the traditionalist and progressives find a lesson/activity/short video that they both agree is lovely teaching.
2) Same thing, but they both agree it’s lousy teaching.
3) Can each identify a whatever that they like, but their sparring partner doesn’t. Can they explain why.
4) If the disagreement persists, can they explain why they think it does?

Elizabeth Statmore uses an explanation protocol called Talking Points and brings student voices into the conversation.

Tracy Zager excerpts quotes from mathematicians on the value of explanation in their own work.

You’ll find another great exchange between Brett Gilland and Katharine Beals (search their names throughout the comments) which ends rather unconventionally for Internet-based discussions of math education.

2015 Nov 28 Education Realist has posted a response that dives into the difference between elementary math ed (the site of Garelick & Beal’s research) and middle school math ed (the site of Garlelick & Beal’s arguments).